8 DEPENDENCE OF THE SOLUTION ON INITIAL CONDITIONS 305

(10.8.2) With the notations of (10.8.1), suppose f is p times continuously
differentiable (resp. E is finite dimensional and f is indefinitely differentiable,
resp. E is finite dimensional and f is analytic) in I x H. Then it is possible to
choose J and V such that the function (t, tQ , ;c0) -» u(t, /0 , x0) is p times con-
tinuously differentiable (resp. indefinitely differentiable, resp. analytic) in
J x J x V.

Indeed, if we write v(s, t0 , XQ) = u(tQ + s, t0 , XQ) — x0 , we see that
5- -» v(s9 tQ , x0) is a solution of the equation

which takes the value 0 at the point s = Q; the result then follows from
(10.7.4) and (10.7.5).

For linear differential equations, there are much more precise results. The
equation

(10.8.3) x' = A(t)-x

is called the homogeneous linear dijfferential equation associated to (10.6.2);
the difference of any two solutions of (10.6.2) in I is a solution of (10.8.3)
in I, and the solutions of (10.8.3) in I constitute a vector subspace ffl of
the space #E(I) of all continuous mappings of I into E.

(10.8.4) For each (s9 x0), let t -> u(t9 s, XQ) be the unique solution of (10.8.3)
defined in I and such that u(s, s, XQ) = x0.

(1) For each t e I, the mapping x0 -»u(t, s, XQ) is a linear homeomorphism
C(t9 s) e J?(E) ofE onto itself.

(2) The mapping t -> C(f, s) of I into the Banach space J5f (E) is equal to
the solution of the linear homogeneous differential equation

(10.8.4.1) U' = A(t)° U

which is equal to 7E (identity mapping of E) for t = s.

(3) For any three points r, s, t in I

(10.8.4.2) C(r, 0 = C(r, s) - C(s, t) and C(s, t) = (C(/, ^))""a.

It is clear that u(t, s, jcj + u(t9 s9 x2) (resp. lu(t, s, *0)) is a solution of
(10.8.3) which is equal to jq + x2 (resp. AJCO) for t — s; hence (10.6.4) it is
equal to u(t, s, x1 + x2) (resp. «/(/, s, Ax0)) in I, which proves that the
mapping xQ-+u(t, s, x0) is linear; let us write it C(f, s) (we have not yet
proved that this mapping is continuous in E). any s e J, and in particular,